(One in a series of six articles on Mathematics and Medicine
being distributed by the Joint Policy Board for Mathematics
in celebration of Mathematics Awareness Week 1994.)

Magellan owed a lot to the compass, a simple device that helped the
sailors of the Old World navigate safely across the vast oceans of
the earth. What if scientists had a compass for exploring the
mysterious river of human consciousness? University of California-
San Francisco researcher Dana Redington thinks that a branch of
mathematics called nonlinear dynamics may provide scientists with
such a tool.

Redington is a psychophysiologist -- a scientist who studies the
nature of the connection between mind and body. He and research
partner Steven Reidbord, a psychiatrist, are using the mathematical
tools of nonlinear dynamics to analyze the way people's heart rates
change when they talk to their therapists.

Everyday experience tells us that the heart and mind are connected.
In the classic "fight-or-flight" reaction, a sudden threat or flash
of anger sends a squirt of adrenaline into the blood, causing the
heart to pound.

Redington and Reidbord are looking for something less obvious than
that, however. They believe that because of the way the mind and
body are wired together, what people think and feel can affect the
complexity of the heart rate -- how much it slows down or speeds up
over a period of time. The scientists record this information
using an electronic heart monitoring machine.

The problem is that these heart-rate changes are subtle and
complex, and can be overlooked easily. This is why Redington and
Reidbord turned to the mathematical area of nonlinear dynamics,
which deals with very complex natural processes. "The thing that's
so compelling about nonlinear dynamics is that extremely
complicated forms of behavior can be described by very simple,
almost elegant equations," Redington explains.

Explaining what nonlinear dynamics is all about can be difficult.
One half of the term is relatively simple, though: dynamics is the
study of motion. Mathematically, it doesn't matter what kind of
motion, whether a swinging pendulum or the way a person's heart
rate slows down or speeds up.

The nonlinear part is harder to define. Jerry Bona, head of the
mathematics department at Pennsylvania State University, says that
the best way to define nonlinear dynamics is by saying what it's
not. A complicated linear process, he explains, can be described
mathematically by breaking it down into simpler parts, analyzing
each one separately, and then adding all the answers up. In
principle, this is like the assembly-line system used to build
automobiles. But to understand a nonlinear system, you can't do
this. Mathematically, you have to look at the whole system all at
once.

Redington and many other scientists view the mind as a nonlinear
system. Part of that nonlinearity stems from the fact that the
brain is connected through the nervous system to the other organs
of the body -- and vice versa. The brain contains a number of
interconnected circuits, or clocks, Redington explains. Each
regulates a different bodily process or organ: breathing, heart
rate, and digestion, for example. These clocks are connected to
each other and to the wiring of higher brain functions such as
thought and emotion.

When everything is working right, these processes all work together
to minimize wasted energy. For example, your heart beat changes
slightly when you inhale to compensate for the greater pressure in
your chest cavity. "Under healthy circumstances, all the organs of
the body are working together in a nice, synchronized dance,"
Redington explains, "but when you get into a disease state, like
stress, the dance becomes lopsided and unbalanced."

Redington and Reidbord are trying to show that these shifts in one
of the brain's internal clocks -- heart rate -- reflect our
shifting mental states. They define these mental states as
distinctive modes of thought, ranging from meditative calm to
unmitigated panic. "We're trying to use physiology -- the way
patterns in the body occur in time -- to give us more information
about what's going on in a person's head," Redington explains.

This marriage of mathematics and psychology might someday better
explain the long-suspected links between mind and health. But
strictly in terms of science, says Redington, nonlinear dynamics is
just a good tool for exploring human consciousness. "When you look
through a telescope, you start to see in much finer detail, and
you're better able to describe things," Redington explains.
"That's exactly what I think nonlinear dynamics is all about."